Ratio and Proportion Exercises with Word Problems and Solutions

Ratio and proportion are fundamental concepts in mathematics that are used to compare quantities and solve real-life problems involving relationships between numbers. These concepts appear frequently in everyday situations. Regular practice helps build accuracy, speed, and confidence. This section offers a mix of ratio and proportion exercises, including MCQs, word problems, and practical questions to strengthen your understanding.

Table of Contents:

• What Is a Ratio?
• What Is Proportion?
• Types of Proportion
• Important Formulas of Ratio and Proportion
• Exercise Set 1: Basic Ratio Exercises
• Exercise Set 2: Proportion Exercises
• Exercise Set 3: Direct and Inverse Proportion
• Exercise Set 4: Ratio Word Problems
• Exercise Set 5: Advanced Problems
• Exercise Set 6: Real-Life Application Exercise
• Common Mistakes to Avoid
  
  

What Is a Ratio?

A ratio is a way of comparing two quantities of the same kind.

For example, if a class has 20 boys and 30 girls, the ratio of boys to girls is written as 20:30, which simplifies to 2: 3.

Key points to remember:

• Ratios are always between quantities of the same unit

• A ratio a:b can also be written as the fraction a/b

• A ratio is said to be in its simplest form when the HCF of both terms is 1

• The first term is the antecedent; the second is the consequent.
  
  

What Is Proportion?

A proportion is an equation that states two ratios are equal. If a:b = c:d, we say that a, b, c, d are in proportion.

This is written as: a:b :: c:d, which means a × d = b × c (cross-multiplication rule)

Here:

• a and d are called the extremes

• b and c are called the means

• The rule is: Product of extremes = Product of means
  
  

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Types of Proportion

1. Direct Proportion: When one quantity increases, the other increases in the same ratio (and vice versa).

Formula: x₁/y₁ = x₂/y₂

Real-life example: If 5 pens cost ₹40, what will 8 pens cost?

5/40 = 8/y

y = (8 × 40)/5 = ₹64

2.  Inverse Proportion: When one quantity increases, the other decreases proportionally.

Formula: x₁ × y₁ = x₂ × y₂

Real-life example: 4 workers complete a task in 12 days. How many days will 6 workers take?

4 × 12 = 6 × d

d = 48/6 = 8 days

3. Continued Proportion: Three quantities a, b, c are in continued proportion when:

a:b = b:c ⇒ b² = a × c

Here b is called the mean proportional between a and c.

Important Formulas of Ratio and Proportion

Exercise Set 1: Basic Ratio Exercises

Exercise 1.1: Express the following in simplest form:

(a) 64:48

(b) 150:250

(c) 1.2:1.8

Solution: a) 64:48

HCF(64, 48) = 16

Dividing each term of 64:48 by 16, we get 4:3

Therefore, the simplest form of 64:48 is 4:3

(b) 150:250

HCF (150, 250) = 50

Dividing each term of 150:250 by 50, we get 3:5

Therefore, the simplest form of 150:250 is 3:5

(c) 1.2:1.8

Multiply both by 10 ⇒ 12:18

HCF(12,18) = 6

Dividing each term of 12:18 by 6, we get 2:3

Therefore, the simplest form of 1.2:1.8 is 2:3.

Exercise 1.2: In a school, there are 240 girls and 180 boys. Find:

(a) Ratio of girls to boys

(b) Ratio of boys to total students

Solution:

Total students = 240 + 180 = 420

(a) Girls:Boys = 240:180 = 4:3

(b) Boys:Total = 180:420 = 3:7

Exercise 1.3: Two numbers are in the ratio 5: 3. If their sum is 64, find the numbers.

Solution: Let the numbers be 5k and 3k.

5k + 3k = 64

8k = 64 ⇒ k = 8

Numbers = 5 × 8 = 40 and 3 × 8 = 24

Therefore the required numbers are 40 and 24

Exercise 1.4: Divide ₹720 among A, B, and C in the ratio 2:3:4.

Solution: Sum of parts = 2 + 3 + 4 = 9

A's share = (2/9) × 720 = ₹160

B's share = (3/9) × 720 = ₹240

C's share = (4/9) × 720 = ₹320

Exercise 1.5: The ratio of mangoes to apples in a basket is 4:5. If there are 36 mangoes, how many apples are there?

Solution: Given, mangoes: apples = 4:5 and number of mangoes = 36

4/5 = 36/x

x = (36 × 5)/4 = 45 apples

Hence, the number of apples = 45.

Exercise Set 2: Proportion Exercises

Exercise 2.1: Find the fourth proportional to 5, 10, and 15.

Solution: Let the fourth proportional be x.

5:10 :: 15:x

⇒ 5x = 10 × 15 = 150

⇒ x = 30

Exercise 2.2: Find the mean proportional between 9 and 25.

Solution: Mean proportional = √(9 × 25) = √225 = 15

Exercise 2.3: Are 4, 12, 5 and 15 in proportion?

Solution:

Product of extremes = 4 × 15 = 60

Product of means = 12 × 5 = 60

Both equal
Therefore, 4, 12, 5 and 15 are in proportion.

Exercise 2.4: If 15:18 :: x:24, find x.

Solution: 18x = 15 × 24 = 360

⇒ x = 20

Exercise 2.5: Find the third proportional to 6 and 12.

Solution: 6:12 :: 12:x

⇒ 6x = 144

⇒ x = 24

Exercise Set 3: Direct and Inverse Proportion

Exercise 3.1: A car travels 180 km in 3 hours. How far will it travel in 5 hours at the same speed?

Solution: Given, Distance travelled by car = 180 km

Time taken by car to travel 180 km = 3 hrs.

Distance is directly proportional to time.

180/3 = x/5

x = (180 × 5)/3 = 300 km

Therefore, the car travels 300 km in 5 hrs.

Exercise 3.2: If 12 workers can lay bricks for a wall in 8 days, how many workers are needed to do the same work in 6 days?

Solution: Given 12 workers can lay bricks for a wall in 8 days.

Total work = 12 × 8 = x × 6

x = 96/6 = 16 

Therefore, the required number of workers to complete the work in 6 days is 16 workers.

Exercise 3.3: A cistern is filled by 3 pipes in 4 hours. If 2 more pipes (of the same capacity) are added, in how many hours will the cistern be filled?

Solution: Given the cistern is filled by 3 pipes in 4 hours. 

Total work done = 3 pipes × 4 hours = 5 pipes × x hours

x = 12/5 = 2.4 hours (2 hours 24 minutes)

Hence, if two more pipes are added, the cistern will be filled in 2 hours 24 minutes.

Exercise 3.4: If 8 bags of wheat cost ₹1,200, what is the cost of 15 bags?

Solution: Given, cost of 8 bags of wheat =  ₹1,200

8/1200 = 15/x

x = (1200 × 15)/8 = ₹2,250

Therefore, the cost of 15 bags of wheat is ₹2,250.

Exercise 3.5: A train covers a certain distance in 2 hours 30 minutes at 90 km/h. How long will it take at 75 km/h?

Solution: Speed and time are inversely proportional.

90 × 2.5 = 75 × t

t = 225/75 = 3 hours

The train will cover the same distance in 3 hours at the speed of 75 km/h.

Exercise Set 4: Ratio Word Problems

Exercise 4.1: The salaries of A and B together amount to ₹50,000. A spends 80% of his salary, and B spends 70% of his salary. If their savings are in the ratio 4:3, find each person's salary.

Solution: Let A's salary =  a, B's salary =  b

a + b = 50,000

A's savings = 20%  of a = 0.20a; B's savings = 30%  of b = 0.30b

0.20a / 0.30b = 4/3

⇒ 0.60a = 1.20b

⇒ a = 2b

Substituting: 2b + b = 50,000 ⇒  b = ₹16,667 (approx.)

Therefore, Salary of A = ₹33,333; Salary of B = ₹16,667

Exercise 4.2: A mixture contains milk and water in the ratio 5:2. If 14 litres of water is added, the ratio becomes 5: 4. Find the original quantity of milk.

Solution: Let original quantities be milk = 5k and water = 2k

After adding 14 L water: 5k/(2k+14) = 5/4

⇒ 20k = 10k + 70

⇒ 10k = 70
⇒ k = 7

Original quantity of milk = 5 × 7 = 35 litres

Exercise 4.3: Two numbers are in the ratio 3: 5. If 8 is subtracted from each, they are in the ratio 1:3. Find the numbers.

Solution: Let the numbers be 3k and 5k.

(3k − 8)/(5k − 8) = 1/3

⇒ 9k − 24 = 5k − 8

⇒ 4k = 16

⇒ k = 4

The required numbers are 12 and 20

Exercise 4.4: Three partners, A, B and C invest capital in the ratio 5: 6: 9. After a year, the profit is ₹60,000. Find each partner's share.

Solution: Sum = 5 + 6 + 9 = 20

A’s share = (5/20) × 60,000 = ₹15,000

B’s share = (6/20) × 60,000 = ₹18,000

C’s share = (9/20) × 60,000 = ₹27,000

Exercise 4.5: If a:b = 3:4 and b:c = 5:6, find a: c.

Solution: Given  a:b = 3:4 and b:c = 5:6

a:c = (a/b) × (b/c) = (3/4) × (5/6) = 15/24 = 5:8

Exercise Set 5: Advanced Problems

Exercise 5.1: A and B together have ₹1,210. If 4/15 of A's amount equals 2/5 of B's amount, how much does B have?

Solution: Given A and B together have ₹1,210.

(4/15)A = (2/5)B

A = (2/5) × (15/4) × B = (3/2)B

⇒ A:B = 3:2

⇒ B's share = (2/5) × 1,210 = ₹484

Exercise 5.2: In a bag, there are coins of 50p, 25p, and 10p in the ratio 2:5:3, totalling ₹510. Find the number of coins of each type.

Solution: Let common ratio be 100k.

Let the number of coins be 200k (50p), 500k (25p) and 300k (10p)

Value = 100k + 125k + 30k = 255k = 510

⇒ k = 2

⇒ 50p coins = 200×2 = 400

⇒ 25p coins = 500×2 = 1000

⇒ 10p coins = 300×2 = 600

Exercise 5.3: A mixture of sugar solution and coloured water is in the ratio 4: 3. When 10 litres of coloured water is added, the ratio becomes 4: 5. Find the initial quantity of sugar solution.

Solution: Let Sugar = 4k, water = 3k

4k/(3k + 10) = 4/5

⇒ 20k = 12k + 40

⇒ 8k = 40

⇒ k = 5

Sugar solution = 4 × 5 = 20 litres

Exercise 5.4: A:B = 2:3, B:C = 4: 5. What is A:B:C?

Solution: Given A:B = 2:3, B:C = 4:5.

Make B equal: A:B = 8:12, B:C = 12:15

A:B:C = 8:12:15

Exercise 5.5: The ratio of the ages of two persons is 4: 7. Eight years ago, the ratio was 1:2. Find their present ages.

Solution: Let present ages of two persons be  4k and 7k

(4k − 8)/(7k − 8) = 1/2

⇒ 8k − 16 = 7k − 8

⇒ k = 8

Therefore, the required ages are 32 years and 56 years.

Exercise Set 6: Real-Life Application Exercise

Exercise 6.1 (Recipe Scaling): A cake recipe for 4 people requires 200g flour, 100g butter, and 150g sugar. How much of each ingredient is needed for 10 people?

Solution: Given the cake recipe for 4 people requires 200g flour, 100g butter, and 150g sugar.

Flour = (200 × 10)/4 = 500g

Butter = (100 × 10)/4 = 250g

Sugar = (150 × 10)/4 = 375g

Therefore, for a cake recipe for 10 people the required amount of flour is 500g, butter = 250g and sugar = 375g

Exercise 6.2 (Map Reading): On a map, 2 cm represents 50 km. If the distance between two cities on the map is 7.5 cm, what is the actual distance?

Solution: 2/50 = 7.5/x

x = (7.5 × 50)/2 = 187.5 km

The actual distance between the two cities is 187.5 km.

Exercise 6.3 (Speed-Distance): A scooter travels 330 km in 6 hours. How many kilometres will it cover in 9 hours at the same speed?

Solution: Given the scooter travels 330 km in 6 hours.

330/6 = x/9

x = (330 × 9)/6 = 495 km.

The scooter travels 495 km in 9 hours with the same speed.

Exercise 6.4 (Workforce Planning): A construction project can be completed by 18 workers in 24 days. If the project needs to be finished in 16 days, how many workers are needed?

Solution: Given the construction project can be completed by 18 workers in 24 days.

Total work = 18 × 24 = x × 16

x = 432/16 = 27 workers

Therefore, 27 workers are required to finish the work in 16 days.

Common Mistakes to Avoid

• Confusing order in ratios: The ratio a:b ≠ b:a. If boys:girls = 3:5, it does NOT mean girls:boys = 3:5.

• Not simplifying before comparing: Always bring ratios to their simplest form before deciding if they're equal.

• Wrong identification of proportion type: Always ask, if one value goes up, does the other go up or down? Up→Up = direct; Up→Down = inverse.

• Adding ratios directly: You cannot add 2:3 + 4:5 directly. Convert to fractions first.

• Ignoring units: Ratios only work between quantities of the same unit. Always convert first if needed.